AT 620 Notes These notes were prepared by Prof. Steven A. Rutledge (and adapted slightly for the Fall 2009 course, and again slightly for this year) You may access Prof. Cotton s notes, password cloud9 http://rams.atmos.colostate.edu/at620old/at620notes.html DO NOT DISTRIBUTE OUTSIDE CSU 1
Motivation: understanding energy and its transformations in the atmosphere 3 T outflow 2 4 1 T sea surface Hurricanes are giant engines that convert heat into wind energy http://apollo.lsc.vsc.edu/classes/met130/notes/chapter15/vertical_circ.html Understanding the phase changes of water and associated energy changes 2
Motivation, continued Thermodynamics provides a basis for understanding nucleation, the process that forms small water droplets and ice crystals, and thus is the foundation for the study of cloud and precipitation processes. Particles (aerosols) and Nucleation of Water and Ice Aerosol indirect effects on climate (those occurring through the link to clouds and precipitation) 3
Thermodynamics is the study of equilibrium states We can use thermodynamics to predict the direction of transitions, but NOT the rate Kinetics, a separate field of study We observe equilibrium states via primitive variables those that can be measured Temperature, pressure, composition But equilibrium (and the laws governing it) is really about energy, expressed in terms of derived properties ( enthalpy, entropy, ) 4
Lecture 1: quick overview of topics Open vs. closed system What does adiabatic mean? Intensive vs. extensive properties What does specific mean (e.g., specific volume )? Ideal Gas Law Equilibria: 1. Thermal 2. Mechanical 3. Chemical And: stable, unstable, metastable State function vs. work and heat 5
REVIEW AND BASIC CONCEPTS a. Definitions Define a system (often can be chosen for convenience) A portion of matter, whose study focuses on its properties, interaction and evolution, from a thermodynamic point of view. That part of the universe under consideration (Wikipedia) All the material environment which may eventually interact with the system is known as the surroundings OPEN System exchange energy (heat and work) and matter with surroundings CLOSED System can exchange energy, but no exchange of matter with surroundings A system is described by its properties or physical variables that express these properties. 6
A more stringent definition of a closed system (isolated system) is that there is no exchange of energy, volume or chemical constituents with the surroundings. For a closed system: mass & chemical composition define the system. properties like temperature & pressure define the system s state Define EXTENSIVE PROPERTIES Take two identical systems each having volume V and chemical constituents given by N 1, N 2, N r where N represents the mole number Now merge these two systems. New volume = 2 X V N new = 2 X N old 7
Parameters that have values in a composite system equal to the sum of the values in each subsystem are called Extensive Parameters. Extensive Variables depend on the size of the system! Intensive Property: does not depend on mass (or size) Intensive and extensive properties are usually denoted by lowercase and uppercase symbols, respectively. p, T- Intensive properties V- Extensive properties Defining an intensive property from an extensive property; Example: is referred to as a specific property Specific Volume To convert an extensive variable into an intensive variable, simply divide by mass V, m are Extensive Variables density is an Intensive Variable 8
More Definitions: Homogeneous System- every intensive variable (e.g. T) has the same value at every point in the system. EXAMPLE: small air parcel if just gas phase. Heterogeneous System- A collection of homogeneous systems, each different from one another. Each homogeneous system is referred to as a phase. Properties of extensive variables for a heterogeneous system; e.g. mass in phase specific value of extensive variable Internal Energy Inhomogeneous system- a system in which the intensive variables change in a continuous manner from one point to another. Atmosphere: T decreases linearly with height. 9
More Definitions: Homogeneous System- every intensive variable (e.g. T) has the same value at every point in the system. EXAMPLE: small air parcel if just gas phase. Heterogeneous System- A collection of homogeneous systems, each different from one another. Each homogeneous system is referred to as a phase. Properties of extensive variables for a heterogeneous system; e.g. mass in phase specific value of extensive variable Internal Energy This gradient would imply energy transfer a RATE of change Inhomogeneous system- a system in which the intensive variables change in a continuous manner from one point to another. Atmosphere: T decreases linearly with height. 10
Equilibrium (brief look; details later) Most of our work in thermodynamics of the atmosphere will be dealing with thermodynamic systems moving from an unstable state to a stable state. Equilibrium: The state of a system remaining constant over time (an ok definition for now); a minimum in energy (or maximum in entropy) UNSTABLE equilibrium: a small change in the state of a system causes large departures from the original state. RH crit RH EXAMPLE: small droplet within a field of water vapor R crit r -small changes in drop size lead to changes in the droplet vapor pressure, relative to the environment -what are accompanying changes in total energy? METASTABLE EQUILIBRIUM Example: supercooled droplet local minimum, not global minimum; stable to small changes in temperature & pressure; sudden freezing if droplet contacts freezing nucleus 11
A system in stable equilibrium is not only invariant in time, but small changes in the intensive properties of the system always finds the system referring to its equilibrium position (state). There are 3 components to equilibrium we will consider: THERMAL ( T ) MECHANICAL ( p ) CHEMICAL ( µ, chemical potential; related to Gibbs Free Energy) -> we can sometimes use vapor pressure here Aside: we assume the environment seeks equilibrium. We often formulate models this way, where the rate of change is proportional to the departure from equilibrium. Example: latent heat transfer at sea-air interface is modelled by Saturation specific humidity of water (ocean) Saturation specific humidity in air (boundary layer) 12
TEMPERATURE A Consider the following system: B adiabatic wall- no heat flow diathermic wall- heat flow Wait for a period of time: A&B are said to be in thermal equilibrium. All bodies in thermal equilibrium have the same temperature. Temperature is defined relative to a reference body that attains thermal equilibrium with the substance (body). (This reference body is of course a thermometer, which is any device with a thermometric substance that can be related to the thermal state of the body) Substance Gas, constant volume Gas, constant pressure Thermocouple Platinum wire Hg, const p Property X Pressure Specific volume EMF (voltage) Resistance Specific volume = a property of a thermometric substance 13
System of Units: adopt SI/MKS Kelvin scale of temperature is defined by assigning the number 273.16 to the temperature of a mixture of pure ice, water and water vapor in equilibrium. Acceleration (g) m s -2 9.8 m s -2 Density ( ρ ) kg m -3 Force (N, Newton) kg m s -2 Pressure (Pa, pascal) N m -2 =kg m -1 s -2 1 Pa= 10 dynes/ cm 2 = 10 baryes= 10-5 bars 1 mb= 10 2 Pa (surface pressure; 1013 mb or about 10 5 Pa) Energy (J Joule) kg m 2 s -2 1J=10 7 ergs=0.2389 calories Specific energy J kg -1 ; m 2 s -2 Calorie energy required to raise temperature of 1g of H 2 O at 15 C by 1 K 1 calorie= 4.1868 J 14
Another important unit/term is the MOLE (mol) The amount of a substance which contains as many elementary units as there are C atoms in 12 g of 12 C. A mole of any substance contains the same number of molecules, which is N A (Avogadro s number) N A = 6.023 X 10 23 molecules / mole One (1) atomic weight of any element contains N A particles. 15
EQUATION OF STATE ( GAS LAW ) Relationships that express intensive parameters (e.g., p, T) in terms of extensive parameters are called equations of state. Early lab studies determined that p, V, and T could be related by an equation of state. Trivia: Boyle s Law, 1662! (P and V inversely proportional) Charles Law, 1787: V proportional to T Avogadro s Law, 1811: equal volumes of ideal gases have same # of molecules Ideal Gas Law: from combining gas laws, proposed by Clapeyron, 1834 IDEAL GAS LAW: Equation of state for ideal gases (gases where molecules have no interactions; no molecular attractive forces; and molecules are assumed to occupy zero volume). pv= nr*t= mr*t/m= mrt n= number of moles= m/m; m = mass, g; M= molecular weight, g/mole R* = universal gas constant, 8.3143 J mol -1 K -1 R = R*/M, specific gas constant, J g -1 K -1 16
p, V and T are often referred to as STATE VARIABLES Introducing density, The SPECIFIC VOLUME is the inverse of density: The universal gas constant follows from Avogadro s hypothesis, which says, gases containing the same number of molecules occupy the same volume at the same p and T and Ideal Gas law can also be written in the form for individual molecules using Boltzmann s constant 17
Check the units for the gas constants: units P V = n R* T But we had a value of Does it work out? ----------------------- Recall Force (N, Newton) kg m s -2 Pressure (Pa, pascal) N m -2 =kg m -1 s -2 Energy (J, Joule) kg m 2 s -2 18
Check the units for the gas constants: units, continued P V = m R T We found a value for DRY AIR (MW ~ 29 g mol -1 ) of Does it work out? ----------------------- Recall Force (N, Newton) kg m s -2 Pressure (Pa, pascal) N m -2 =kg m -1 s -2 Energy (J, Joule) kg m 2 s -2 19
Aside: REAL GASES Behavior of real gases (molecular attractive forces; molecules have non-zero volume) can be approximated by the Van der Waals equation (other equations also proposed): For 1 mole ν =specific volume Follows from a model of point molecules moving independently from one another Two simple corrections were developed by Van der Waals 1. Molecules have non-zero volume or n=# of moles nb is volume occupied by molecules V=total volume a,b constants for each particular gas 2. Pressure is reduced by ; decrease of pressure is proportional to number of interacting pairs of molecules per unit volume Molecule approaching wall experiences a net backward attraction due to remaining molecules in volume; thus pressure exerted on wall is reduced. (See notes at end for more) 20
Mixture of IDEAL GASES Dalton s Laws of partial pressures In a mixture of gases, the pressure P i of the i th constituent is defined as the pressure that the gas would exert if the i th gas occupied the volume alone. Total pressure For each gas, with Applying Dalton s Law: Since our goal is to represent properties of a mixture of ideal gases as those of a single gas, we define a mean gas constant and molecular weight for the mixture as Gas constant Molecular weight M air ~ 29 g mole -1 (useful) Hence IDEAL GAS LAW FOR A MIXTURE OF GASES (LIKE OUR ATMOSPHERE) 21
Mole fraction for each constituent: For 2 gases Of course (sum of mole fractions is equal to 1) Atmospheric Composition Mixture of Ideal gases Water in 3 phases Liquid Aerosols Solid Cloud Formation Atmospheric Chemistry Radiation Optics Climate Atmospheric Electricity 22
TABLE I-4 Main components of dry atmospheric air Gas Molecular Weight Molar (or volume) fraction Mass Fraction (J kg -1 K -1 ) Specific gas constant (J kg -1 K -1 ) (Jkg -1 K -1 ) Nitrogen (N 2 ) 28.013 0.7809 0.7552 296.80 224.15 Oxygen (O 2 ) 31.999 0.2095 0.2315 259.83 60.15 Argon (Ar) 39.948 0.0093 0.0128 208.13 2.66 Carbon dioxide (CO 2 ) 44.010 0.0003 0.0005 188.92 0.09 1.0000 1.0000 H 2 O (v) 18.02 g/mole Volume Fraction 0-0.04 23
TABLE I-5 Minor gas components of atmospheric air. Gas Molar(or volume) Fraction Neon (Ne) 1.8x10-5 Helium (HE) 5.2x10-6 Methane (CH 4 ) 1.5x10-6 Krypton (Kr) 1.1x10-6 Hydrogen (H 2 ) 5.0x10-7 Nitrous oxide (N 2 O) 2.5x10-7 Carbon monoxide (CO) 1.0x10-7 Xeon (Xe) 8.6x10-8 Ozone (O 3 ) Variable. Up to 10-5 in stratosphere. Sulfur dioxide (SO 2 ) Hydrogen sulfide and other reduced sulfur compounds H 2 S, ect.) Nitric oxide (NO) Variable, under 10-8 Nitrogen dioxide (NO 2 ) Ammonia (NH 3 ) Formaldehyde (CH 2 O) 24
TABLE 4.1 Fractional Composition of Planetary Atmospheres Earth Venus Mars Titan TRACE GASES; BUT IMPORTANT! Pressure (mbar) Surface 1013 92,000 6 1496 Cloud Top - ~200 - ~ 10 Major Gases, % CO 2, (CH 4 ) 0.03 b 96.5 95.3 (2-10) N 2 78 3.5 2.7 65-98 O 2 21 <0.1ppm 0.13 - H 2 O 0-2 c 100ppm 100ppm - Ar 0.9 70ppm 1.6 0-25? Trace Gases (ppm) H 2 0.5? ~ 10 2000 He 5 ~ 12 - - CO 0.2 50 700 ~ 100 NO 0.0005 - ~ 3 - O 3 0.4 - ~ 0.01 - HCl - 0.6 - - HF - 0.005 - - Ne 18.2 7 2.5 - Kr 1.14 1 0.3 - Xe 0.087 < 0.1 0.08-25
Tables I-4 and I-5 show 4 major constituents and numerous minor constituents for Earth s Atmosphere. These gases compose dry air or moist air if H 2 O is included. These gases exist in these proportions in roughly the lowest 100km of our atmosphere due to turbulent mixing: 100-120km Diffusive equilibrium occurs above the turbo pause (homopause) Turbulent mixing in the homosphere In the absence of other forces acting on them, gases diffuse at different rates according to their individual molecular weights From solving hydrostatic balance equation (see next slide) or p H=SCALE HEIGHT Above ~100 km, heavier gases (large M) have smaller scale heights e folding distance is less. For our purposes we well consider dry air as a constant mixture which behaves like a pure gas. 26 GAS CONSTANT
Hydrostatic balance (http://eesc.columbia.edu/courses/ees/climate/lectures/atm_phys.html) The atmosphere does not collapse under the downward pull of gravity because of the energy embedded in the movement of the air molecules. This movement creates the force of pressure which counters the gravitational pull on the atmosphere. The balance between the force of pressure and gravity is the hydrostatic balance. To find the expression for the hydrostatic balance, we first note that atmospheric surface pressure is due to the weight of the entire atmospheric column above. As we ascend, there is less of an atmosphere above us, and hence the pressure drops. Consider a column of gas Δz meters tall suspended somewhere in the atmosphere. The reason this column of air does not "fall" down under the pull of gravity is because the pressure acting on its bottom surface is higher than the pressure acting on its top surface. The pressure difference Δp exactly balances the weight (per unit area) of the column. Stated in mathematical terms this balance is written as: Δp = - ρ g Δz Write density in terms of T and p to get in form that yields the scale height where g is the acceleration due to gravity = 9.8 m/s 2. 27
J kg -1 k -1 g mole -1 Useful number! The standard atmosphere (see depiction to the right, from Wikipedia) indicates an average scale height below ~100 km of ~7.64 km, consistent with an average air temperature over that range of close to 260 K Note that equation can be rearranged as At higher altitudes each gas tends to have its own scale height (gravity plays more of a role than mixing)
THE FIRST LAW of THERMODYNAMICS: Conservation of energy Consider a body of molecules v Macroscopic velocity=v Body has 3 types of energies: macroscopic kinetic energy macroscopic potential energy microscopic internal energy, KE and PE of molecules Potential energy (PE) at the microscopic level is associated with molecular attractive forces Kinetic energy (KE) at the microscopic level is associated with molecular motions which are of course proportional to the temperature of the gas. First Law is concerned with conservation of microscopic energy 29
WORK (OF EXPANSION) Atmosphere is a compressible gas!! One type of work we will consider in our study of atmospheric thermodynamics is the work of expansion (e.g., an expanding gas ). If the gas expands,. If the gas contracts,. Formulation of work term (mechanical work): p,v piston displacement dl cross-sectional area A Frictionless Piston Work = area under P-V curve For work done in a cycle, p 1 a 30 2 V
More on Van der Waal s Equation For n=1 mole For large V, V»nb and p» so we see that Van der Waal s Equation takes on the same form as the ideal gas law. Examine pressure reduction term Consider outermost layer of molecules, and layer of molecules just below this layer. There is an attractive force between these 2 layers. Molecules at other positions are subject to a net-zero force. So pressure reduction term is like Point attractive forces between molecules will be proportional to, where ; molecular density Since n=# of moles N A =Avogadro s # We have 31
Volume reduction term Consider atoms or molecules as spheres with diameter D:. for molecules Molecular sphere of influence is (no other molecule exists within diameter D of another molecule, or volume of ) Total volume occupied by N molecules is Or since, Volume reduction term is or per mole This term, would be b in the volume reduction term. So volume reduction is 32
Intermolecular forces Van der Waals forces Intermolecular forces can be repulsive or attractive In general- F Repulsive Attractive r For r large compared to diameter of the molecule, F is attractive - - + + - - (Correlation in electron positions) Van der Waals force is ELECTRICAL in nature positive nucleus - + - negative electron cloud For F Repulsive (electrostatic repulsion of electron clouds). This process is the reason that a liquid is incompressible. 33